UPPER SEMI-CONTINUITY OF RANDOM ATTRACTORS FOR A NON-AUTONOMOUS DYNAMICAL SYSTEM WITH A WEAK CONVERGENCE CONDITION

doi:10.1007/s10473-020-0403-3

数学物理学报(英文版) ›› 2020, Vol. 40 ›› Issue (4): 921-933.doi: 10.1007/s10473-020-0403-3

UPPER SEMI-CONTINUITY OF RANDOM ATTRACTORS FOR A NON-AUTONOMOUS DYNAMICAL SYSTEM WITH A WEAK CONVERGENCE CONDITION

赵文强¹, 张一进²

1. Chongqing Key Laboratory of Social Economy and Applied Statistics, School of Mathematics and Statistics, Chongqing Technology and Business University, Chongqing 400067, China;
2. Chongqing Key Laboratory of Social Economy and Applied Statistics, School of Science, Chongqing University of Posts and Telecommunications, Chongqing 400065, China

收稿日期:2019-06-16 修回日期:2019-10-30 出版日期:2020-08-25 发布日期:2020-08-21
通讯作者: Wenqiang ZHAO E-mail:gshzhao@sina.com
作者简介:Yijin ZHANG,E-mail:Zhangyj@cqupt.edu.cn
基金资助:
This work was supported by CTBU (KFJJ2018101), CTBU ZDPTTD201909, Chongqing NSF (2019jcyj-msxmX0115) and NSFC (11871122).

UPPER SEMI-CONTINUITY OF RANDOM ATTRACTORS FOR A NON-AUTONOMOUS DYNAMICAL SYSTEM WITH A WEAK CONVERGENCE CONDITION

Wenqiang ZHAO¹, Yijin ZHANG²

1. Chongqing Key Laboratory of Social Economy and Applied Statistics, School of Mathematics and Statistics, Chongqing Technology and Business University, Chongqing 400067, China;
2. Chongqing Key Laboratory of Social Economy and Applied Statistics, School of Science, Chongqing University of Posts and Telecommunications, Chongqing 400065, China

Received:2019-06-16 Revised:2019-10-30 Online:2020-08-25 Published:2020-08-21
Contact: Wenqiang ZHAO E-mail:gshzhao@sina.com
Supported by:
This work was supported by CTBU (KFJJ2018101), CTBU ZDPTTD201909, Chongqing NSF (2019jcyj-msxmX0115) and NSFC (11871122).

摘要/Abstract

摘要： In this paper, we develop the criterion on the upper semi-continuity of random attractors by a weak-to-weak limit replacing the usual norm-to-norm limit. As an application, we obtain the convergence of random attractors for non-autonomous stochastic reaction-diffusion equations on unbounded domains, when the density of stochastic noises approaches zero. The weak convergence of solutions is proved by means of Alaoglu weak compactness theorem. A differentiability condition on nonlinearity is omitted, which implies that the existence conditions for random attractors are sufficient to ensure their upper semi-continuity. These results greatly strengthen the upper semi-continuity notion that has been developed in the literature.

关键词: non-autonomous random dynamical system, random attractor, upper semi-continuity, weak convergence

Abstract: In this paper, we develop the criterion on the upper semi-continuity of random attractors by a weak-to-weak limit replacing the usual norm-to-norm limit. As an application, we obtain the convergence of random attractors for non-autonomous stochastic reaction-diffusion equations on unbounded domains, when the density of stochastic noises approaches zero. The weak convergence of solutions is proved by means of Alaoglu weak compactness theorem. A differentiability condition on nonlinearity is omitted, which implies that the existence conditions for random attractors are sufficient to ensure their upper semi-continuity. These results greatly strengthen the upper semi-continuity notion that has been developed in the literature.

Key words: non-autonomous random dynamical system, random attractor, upper semi-continuity, weak convergence

中图分类号:

35R60

赵文强, 张一进. UPPER SEMI-CONTINUITY OF RANDOM ATTRACTORS FOR A NON-AUTONOMOUS DYNAMICAL SYSTEM WITH A WEAK CONVERGENCE CONDITION[J]. 数学物理学报(英文版), 2020, 40(4): 921-933.

Wenqiang ZHAO, Yijin ZHANG. UPPER SEMI-CONTINUITY OF RANDOM ATTRACTORS FOR A NON-AUTONOMOUS DYNAMICAL SYSTEM WITH A WEAK CONVERGENCE CONDITION[J]. Acta mathematica scientia,Series B, 2020, 40(4): 921-933.

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UPPER SEMI-CONTINUITY OF RANDOM ATTRACTORS FOR A NON-AUTONOMOUS DYNAMICAL SYSTEM WITH A WEAK CONVERGENCE CONDITION

UPPER SEMI-CONTINUITY OF RANDOM ATTRACTORS FOR A NON-AUTONOMOUS DYNAMICAL SYSTEM WITH A WEAK CONVERGENCE CONDITION

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